Mixing
differential forms- exact,closed
$begingroup$
How can I tell if differential forms are exact or closed?
Let's take for example
$$i) quad omega = frac12 dx -frac1y dy -frac2xdz$$
where $x,y,z in mathbb{R}^{+}$ and
$$ii) quad eta = x^2 dy wedge dz +yx dz wedge dx+z^3 dx wedge dy $$
What definition do you use and what does it mean?
Any help very appreciated !
real-analysis differential-forms
edited Jan 9 at 13:02
Mattos
2,73721321
asked Jan 8 at 22:26
constant94constant94
6310
$endgroup$
add a comment |
$begingroup$
How can I tell if differential forms are exact or closed?
Let's take for example
$$i) quad omega = frac12 dx -frac1y dy -frac2xdz$$
where $x,y,z in mathbb{R}^{+}$ and
$$ii) quad eta = x^2 dy wedge dz +yx dz wedge dx+z^3 dx wedge dy $$
What definition do you use and what does it mean?
Any help very appreciated !
real-analysis differential-forms
edited Jan 9 at 13:02
Mattos
2,73721321
asked Jan 8 at 22:26
constant94constant94
6310
$endgroup$
$begingroup$
You compute them...
$endgroup$
– Hawk
Jan 9 at 14:11
add a comment |
$begingroup$
How can I tell if differential forms are exact or closed?
Let's take for example
$$i) quad omega = frac12 dx -frac1y dy -frac2xdz$$
where $x,y,z in mathbb{R}^{+}$ and
$$ii) quad eta = x^2 dy wedge dz +yx dz wedge dx+z^3 dx wedge dy $$
What definition do you use and what does it mean?
Any help very appreciated !
real-analysis differential-forms
edited Jan 9 at 13:02
Mattos
2,73721321
asked Jan 8 at 22:26
constant94constant94
6310
$endgroup$
How can I tell if differential forms are exact or closed?
Let's take for example
$$i) quad omega = frac12 dx -frac1y dy -frac2xdz$$
where $x,y,z in mathbb{R}^{+}$ and
$$ii) quad eta = x^2 dy wedge dz +yx dz wedge dx+z^3 dx wedge dy $$
What definition do you use and what does it mean?
Any help very appreciated !
real-analysis differential-forms
real-analysis differential-forms
edited Jan 9 at 13:02
Mattos
2,73721321
asked Jan 8 at 22:26
constant94constant94
6310
edited Jan 9 at 13:02
Mattos
2,73721321
asked Jan 8 at 22:26
constant94constant94
6310
edited Jan 9 at 13:02
Mattos
2,73721321
edited Jan 9 at 13:02
Mattos
2,73721321
edited Jan 9 at 13:02
Mattos
2,73721321
2,73721321
asked Jan 8 at 22:26
constant94constant94
6310
asked Jan 8 at 22:26
constant94constant94
6310
asked Jan 8 at 22:26
constant94constant94
6310
6310
$begingroup$
You compute them...
$endgroup$
– Hawk
Jan 9 at 14:11
add a comment |
$begingroup$
You compute them...
$endgroup$
– Hawk
Jan 9 at 14:11
$begingroup$
You compute them...
$endgroup$
– Hawk
Jan 9 at 14:11
$begingroup$
You compute them...
$endgroup$
– Hawk
Jan 9 at 14:11
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
A differential form $p$-form $omega$ is closed if $domega = 0$ and exact if there is a $(p-1)$-form $eta$ such that $deta = omega$. From these definitions it can be seen that every exact form is closed, but the converse is not generally true. Now using your definitions for $omega$ and $eta$:
begin{eqnarray}
domega&=&frac{1}{2}d(dx)+frac{1}{y^2}dywedge dy+frac{2}{x^2}dxwedge dz=frac{2}{x^2}dxwedge dz, \
deta &=& 2xdxwedge dy wedge dz +xdywedge dzwedge dx+3z^2dzwedge dxwedge dy=(2x+ 3z^2)dxwedge dywedge dz.
end{eqnarray}
Then $omega$ and $eta$ are not closed.
answered Jan 9 at 13:21
jobejobe
860614
$endgroup$
add a comment |
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
A differential form $p$-form $omega$ is closed if $domega = 0$ and exact if there is a $(p-1)$-form $eta$ such that $deta = omega$. From these definitions it can be seen that every exact form is closed, but the converse is not generally true. Now using your definitions for $omega$ and $eta$:
begin{eqnarray}
domega&=&frac{1}{2}d(dx)+frac{1}{y^2}dywedge dy+frac{2}{x^2}dxwedge dz=frac{2}{x^2}dxwedge dz, \
deta &=& 2xdxwedge dy wedge dz +xdywedge dzwedge dx+3z^2dzwedge dxwedge dy=(2x+ 3z^2)dxwedge dywedge dz.
end{eqnarray}
Then $omega$ and $eta$ are not closed.
answered Jan 9 at 13:21
jobejobe
860614
$endgroup$
add a comment |
$begingroup$
A differential form $p$-form $omega$ is closed if $domega = 0$ and exact if there is a $(p-1)$-form $eta$ such that $deta = omega$. From these definitions it can be seen that every exact form is closed, but the converse is not generally true. Now using your definitions for $omega$ and $eta$:
begin{eqnarray}
domega&=&frac{1}{2}d(dx)+frac{1}{y^2}dywedge dy+frac{2}{x^2}dxwedge dz=frac{2}{x^2}dxwedge dz, \
deta &=& 2xdxwedge dy wedge dz +xdywedge dzwedge dx+3z^2dzwedge dxwedge dy=(2x+ 3z^2)dxwedge dywedge dz.
end{eqnarray}
Then $omega$ and $eta$ are not closed.
answered Jan 9 at 13:21
jobejobe
860614
$endgroup$
add a comment |
$begingroup$
A differential form $p$-form $omega$ is closed if $domega = 0$ and exact if there is a $(p-1)$-form $eta$ such that $deta = omega$. From these definitions it can be seen that every exact form is closed, but the converse is not generally true. Now using your definitions for $omega$ and $eta$:
begin{eqnarray}
domega&=&frac{1}{2}d(dx)+frac{1}{y^2}dywedge dy+frac{2}{x^2}dxwedge dz=frac{2}{x^2}dxwedge dz, \
deta &=& 2xdxwedge dy wedge dz +xdywedge dzwedge dx+3z^2dzwedge dxwedge dy=(2x+ 3z^2)dxwedge dywedge dz.
end{eqnarray}
Then $omega$ and $eta$ are not closed.
answered Jan 9 at 13:21
jobejobe
860614
$endgroup$
A differential form $p$-form $omega$ is closed if $domega = 0$ and exact if there is a $(p-1)$-form $eta$ such that $deta = omega$. From these definitions it can be seen that every exact form is closed, but the converse is not generally true. Now using your definitions for $omega$ and $eta$:
begin{eqnarray}
domega&=&frac{1}{2}d(dx)+frac{1}{y^2}dywedge dy+frac{2}{x^2}dxwedge dz=frac{2}{x^2}dxwedge dz, \
deta &=& 2xdxwedge dy wedge dz +xdywedge dzwedge dx+3z^2dzwedge dxwedge dy=(2x+ 3z^2)dxwedge dywedge dz.
end{eqnarray}
Then $omega$ and $eta$ are not closed.
answered Jan 9 at 13:21
jobejobe
860614
answered Jan 9 at 13:21
jobejobe
860614
answered Jan 9 at 13:21
jobejobe
860614
answered Jan 9 at 13:21
jobejobe
860614
860614
add a comment |
add a comment |
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$begingroup$
You compute them...
$endgroup$
– Hawk
Jan 9 at 14:11